NAG Library Routine Document

H02BBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

+− 5 Parameters

5.1 Description of Printed Output

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

H02BBF solves ‘zero-one’, ‘general’, ‘mixed’ or ‘all’ integer programming problems using a branch and bound method. The routine may also be used to find either the first integer solution or the optimum integer solution. It is not intended for large sparse problems.

2 Specification

SUBROUTINE H02BBF (

ITMAX, MSGLVL, N, M, A, LDA, BL, BU, INTVAR, CVEC, MAXNOD, INTFST, MAXDPT, TOLIV, TOLFES, BIGBND, X, OBJMIP, IWORK, LIWORK, RWORK, LRWORK, IFAIL)

INTEGER	ITMAX, MSGLVL, N, M, LDA, INTVAR(N), MAXNOD, INTFST, MAXDPT, IWORK(LIWORK), LIWORK, LRWORK, IFAIL
REAL (KIND=nag_wp)	A(LDA,*), BL(N+M), BU(N+M), CVEC(N), TOLIV, TOLFES, BIGBND, X(N), OBJMIP, RWORK(LRWORK)

3 Description

H02BBF is capable of solving certain types of integer programming (IP) problems using a branch and bound (B&B) method, see Taha (1987). In order to describe these types of integer programs and to briefly state the B&B method, we define the following linear programming (LP) problem:

Minimize

F (x) = c_{1} x_{1} + c_{2} x_{2} + \dots + c_{n} x_{n}

subject to

\sum_{j = 1}^{n} a_{i j} x_{j} \{\begin{matrix} = \\ \leq \\ \geq \end{matrix}\} b_{i}, i = 1, 2, \dots, m

l_{j} \leq x_{j} \leq u_{j}, j = 1, 2, \dots, n

(1)

If, in (1), it is required that (some or) all the variables take integer values, then the integer program is of type (mixed or) all general IP problem. If additionally, the integer variables are restricted to take only

0

–

1

values (i.e.,

l_{j} = 0

and

u_{j} = 1

) then the integer program is of type (mixed or all) zero-one IP problem.

The B&B method applies directly to these integer programs. The general idea of B&B (for a full description see Dakin (1965) or Mitra (1973)) is to solve the problem without the integral restrictions as an LP problem (first node). If in the optimal solution an integer variable

x_{k}

takes a noninteger value

x_{k}^{*}

, two LP sub-problems are created by branching, imposing

x_{k} \leq [x_{k}^{*}]

and

x_{k} \geq [x_{k}^{*}] + 1

respectively, where

[x_{k}^{*}]

denotes the integer part of

x_{k}^{*}

. This method of branching continues until the first integer solution (bound) is obtained. The hanging nodes are then solved and investigated in order to prove the optimality of the solution. At each node, an LP problem is solved using E04MFF/E04MFA.

4 References

Dakin R J (1965) A tree search algorithm for mixed integer programming problems Comput. J. 8 250–255

Mitra G (1973) Investigation of some branch and bound strategies for the solution of mixed integer linear programs Math. Programming 4 155–170

Taha H A (1987) Operations Research: An Introduction Macmillan, New York

5 Parameters

1: ITMAX – INTEGERInput/Output

On entry: an upper bound on the number of iterations for each LP problem.

On exit: unchanged if on entry

ITMAX > 0

, else

ITMAX = \max (50, 5 \times (N + M))

2: MSGLVL – INTEGERInput

On entry: the amount of printout produced by H02BBF, as indicated below (see Section 5.1 for a description of the printed output). All output is written to the current advisory message unit (as defined by X04ABF).

Value	Definition
0	No output.
1	The final IP solution only.
5	One line of output for each node investigated and the final IP solution.
10	The original LP solution (first node), one line of output for each node investigated and the final IP solution.

3: N – INTEGERInput

On entry:

n

, the number of variables.

Constraint:

N > 0

4: M – INTEGERInput

On entry:

m

, the number of general linear constraints.

Constraint:

M \geq 0

5: A(LDA, $*$ ) – REAL (KIND=nag_wp) arrayInput

Note: the second dimension of the array A must be at least

N

M > 0

and at least

1

M = 0

On entry: the

i

th row of A must contain the coefficients of the

i

th general constraint, for

i = 1, 2, \dots, m

M = 0

then the array A is not referenced.

6: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which H02BBF is called.

Constraint:

LDA \geq \max (1, M)

7: BL( $N + M$ ) – REAL (KIND=nag_wp) arrayInput

8: BU( $N + M$ ) – REAL (KIND=nag_wp) arrayInput

On entry: BL must contain the lower bounds and BU the upper bounds, for all the constraints in the following order. The first

n

elements of each array must contain the bounds on the variables, and the next

m

elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e.,

l_{j} = - \infty

), set

BL (j) \leq - BIGBND

and to specify a nonexistent upper bound (i.e.,

u_{j} = + \infty

), set

BU (j) \geq BIGBND

. To specify the

j

th constraint as an equality, set

BL (j) = BU (j) = β

, say, where

|β| < BIGBND

Constraints:

$BL (j) \leq BU (j)$ , for $j = 1, 2, \dots, N + M$ ;
if $BL (j) = BU (j) = β$ , $|β| < BIGBND$ .

9: INTVAR(N) – INTEGER arrayInput

On entry: indicates which are the integer variables in the problem. For example, if

x_{j}

is an integer variable then

INTVAR (j)

must be set to

1

, and

0

otherwise.

Constraints:

$INTVAR (j) = 0$ or $1$ , for $j = 1, 2, \dots, N$ ;
$INTVAR (j) = 1$ for at least one value of $j$ .

10: CVEC(N) – REAL (KIND=nag_wp) arrayInput

On entry: the coefficients

c_{j}

of the objective function

F (x) = c_{1} x_{1} + c_{2} x_{2} + \dots + c_{n} x_{n}

. The routine attempts to find a minimum of

F (x)

. If a maximum of

F (x)

is desired,

CVEC (j)

should be set to

- c_{j}

, for

j = 1, 2, \dots, n

, so that the routine will find a minimum of

- F (x)

11: MAXNOD – INTEGERInput

On entry: the maximum number of nodes that are to be searched in order to find a solution (optimum integer solution). If

MAXNOD \leq 0

and

INTFST \leq 0

, then the B&B tree search is continued until all the nodes have been investigated.

12: INTFST – INTEGERInput

On entry: specifies whether to terminate the B&B tree search after the first integer solution (if any) is obtained. If

INTFST > 0

then the B&B tree search is terminated at node

k

say, which contains the first integer solution. For

MAXNOD > 0

this applies only if

k \leq MAXNOD

13: MAXDPT – INTEGERInput

On entry: the maximum depth of the B&B tree used for branch and bound.

Suggested value:

MAXDPT = 3 \times N / 2

Constraint:

MAXDPT \geq 2

14: TOLIV – REAL (KIND=nag_wp)Input/Output

On entry: the integer feasibility tolerance; i.e., an integer variable is considered to take an integer value if its violation does not exceed TOLIV. For example, if the integer variable

x_{j}

is near unity then

x_{j}

is considered to be integer only if

(1 - TOLIV) \leq x_{j} \leq (1 + TOLIV)

On exit: unchanged if on entry

TOLIV > 0.0

, else

TOLIV = 10^{- 5}

15: TOLFES – REAL (KIND=nag_wp)Input/Output

On entry: the maximum acceptable absolute violation in each constraint at a ‘feasible’ point (feasibility tolerance); i.e., a constraint is considered satisfied if its violation does not exceed TOLFES.

On exit: unchanged if on entry

TOLFES > 0.0

, else

TOLFES = \sqrt{ε}

(where

ε

is the machine precision).

16: BIGBND – REAL (KIND=nag_wp)Input/Output

On entry: the ‘infinite’ bound size in the definition of the problem constraints. More precisely, any upper bound greater than or equal to BIGBND will be regarded as

+ \infty

and any lower bound less than or equal to

- BIGBND

will be regarded as

- \infty

On exit: unchanged if on entry

BIGBND > 0.0

, else

BIGBND = 10^{20}

17: X(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: an initial estimate of the original LP solution.

On exit: with

IFAIL = 0

, X contains a solution which will be an estimate of either the optimum integer solution or the first integer solution, depending on the value of INTFST. If

IFAIL = 9

, then X contains a solution which will be an estimate of the best integer solution that was obtained by searching MAXNOD nodes.

18: OBJMIP – REAL (KIND=nag_wp)Output

On exit: with

IFAIL = 0

9

, OBJMIP contains the value of the objective function for the IP solution.

19: IWORK(LIWORK) – INTEGER arrayCommunication Array

20: LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which H02BBF is called.

Constraint:

LIWORK \geq (25 + N + M) \times MAXDPT + 5 \times N + M + 4

21: RWORK(LRWORK) – REAL (KIND=nag_wp) arrayCommunication Array

22: LRWORK – INTEGERInput

On entry: the dimension of the array RWORK as declared in the (sub)program from which H02BBF is called.

Constraint:

LRWORK \geq MAXDPT \times (N + 1) + 2 \times {\min (N, M + 1)}^{2} + 14 \times N + 12 \times M

MSGLVL > 0

, the amounts of workspace provided and required (with

MAXDPT = 3 \times N / 2

) are printed. As an alternative to computing MAXDPT, LIWORK and LRWORK from the formulas given above, you may prefer to obtain appropriate values from the output of a preliminary run with the values of MAXDPT, LIWORK and LRWORK set to

1

. If however only LIWORK and LRWORK are set to

1

, then the appropriate values of these parameters for the given value of MAXDPT will be computed and printed unless

MAXDPT < 2

. In both cases H02BBF will then terminate with

IFAIL = 6

23: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if

IFAIL \neq 0

on exit, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

5.1 Description of Printed Output

The level of printed output from H02BBF is controlled by you (see the description of MSGLVL in Section 5).

When

MSGLVL > 0

, the summary printout at the end of execution of H02BBF includes a listing of the status of every variable and constraint. Note that default names are assigned to all variables and constraints. The following describes the printout for each variable.

Varbl	gives the name (V) and index $j$ , for $j = 1, 2, \dots, n$ , of the variable.
State	gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the Feasibility Tolerance, State will be ++ or -- respectively.
Value	is the value of the variable at the final iterate.
Lower Bound	is the lower bound specified for the variable. (None indicates that $BL (j) \leq - BIGBND$ .) Note that if $INTVAR (j) = 1$ , then the printed value of Lower Bound for the $j$ th variable may not be the same as that originally supplied in $BL (j)$ .
Upper Bound	is the upper bound specified for the variable. (None indicates that $BU (j) \geq BIGBND$ .) Note that if $INTVAR (j) = 1$ , then the printed value of Upper Bound for the $j$ th variable may not be the same as that originally supplied in $BU (j)$ .
Lagr Mult	is the value of the Lagrange-multiplier for the associated bound constraint. This will be zero if State is FR or TF. If $x$ is optimal, the multiplier should be non-negative if State is LL, and non-positive if State is UL.
Residual	is the difference between the variable Value and the nearer of its bounds $BL (j)$ and $BU (j)$ .

The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’,

BL (j)

and

BU (j)

are replaced by

BL (n + j)

and

BU (n + j)

respectively, and with the following change in the heading.

L Con

gives the name (L) and index

j

, for

j = 1, 2, \dots, m

, of the constraint.

When

MSGLVL > 1

, the summary printout at the end of every node during the execution of H02BBF is a listing of the outcome of forcing an integer variable with a noninteger value to take a value within its specified lower and upper bounds.

Node No	is the current node number of the B&B tree being investigated.
Parent Node	is the parent node number of the current node.
Obj Value	is the final objective function value. If a node does not have a feasible solution then No Feas Soln is printed instead of the objective function value. If a node whose optimum solution exceeds the best integer solution so far is encountered (i.e., it does not pay to explore the sub-problem any further), then its objective function value is printed together with a CO (Cut Off).
Varbl Chosen	is the index of the integer variable chosen for branching.
Value Before	is the noninteger value of the integer variable chosen.
Lower Bound	is the lower bound value that the integer variable is allowed to take.
Upper Bound	is the upper bound value that the integer variable is allowed to take.
Value After	is the value of the integer variable after the current optimization.
Depth	is the depth of the B&B tree at the current node.

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Note: H02BBF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$: No feasible integer point was found, i.e., it was not possible to satisfy all the integer variables to within the integer feasibility tolerance (determined by TOLIV). Increase TOLIV and rerun H02BBF.

$IFAIL = 2$: The original LP solution appears to be unbounded. This value of IFAIL implies that a step as large as BIGBND would have to be taken in order to continue the algorithm (see Section 8).

$IFAIL = 3$: No feasible point was found for the original LP problem, i.e., it was not possible to satisfy all the constraints to within the feasibility tolerance (determined by TOLFES). If the data for the constraints are accurate only to the absolute precision $σ$ , you should ensure that the value of the feasibility tolerance is greater than $σ$ . For example, if all elements of $A$ are of order unity and are accurate only to three decimal places, the feasibility tolerance should be at least $10^{- 3}$ (see Section 8).

$IFAIL = 4$: The maximum number of iterations (determined by ITMAX) was reached before normal termination occurred for the original LP problem (see Section 8).

$IFAIL = 5$: Not used by this routine.

$IFAIL = 6$: An input parameter is invalid.

$IFAIL = 7$: The IP solution reported is not the optimum IP solution. In other words, the B&B tree search for at least one of the branches had to be terminated since an LP sub-problem in the branch did not have a solution (see Section 8).

$IFAIL = 8$: The maximum depth of the B&B tree used for branch and bound (determined by MAXDPT) is too small. Increase MAXDPT (along with LIWORK and/or LRWORK if appropriate) and rerun H02BBF.

$IFAIL = 9$: The IP solution reported is the best IP solution for the number of nodes (determined by MAXNOD) investigated in the B&B tree.

$IFAIL = 10$: No feasible integer point was found for the number of nodes (determined by MAXNOD) investigated in the B&B tree.

$IFAIL = 11$: Although the workspace sizes are sufficient to meet the documented restriction, they are not sufficiently large to accommodate an internal partition of the workspace that meets the requirements of the problem. Increase the workspace sizes.

$Overflow$: It may be possible to avoid the difficulty by increasing the magnitude of the feasibility tolerance (TOLFES) and rerunning the program. If the message recurs even after this change, see Section 8.

7 Accuracy

H02BBF implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.

8 Further Comments

The original LP problem may not have an optimum solution, i.e., H02BBF terminates with

IFAIL = 2

3

4

or overflow may occur. In this case, you are recommended to relax the integer restrictions of the problem and try to find the optimum LP solution by using E04MFF/E04MFA instead.

In the B&B method, it is possible for an LP sub-problem to terminate without finding a solution. This may occur due to the number of iterations exceeding the maximum allowed. Therefore the B&B tree search for that particular branch cannot be continued. Thus the returned solution may not be optimal. (

IFAIL = 7

). For the second and unlikely case, a solution could not be found despite a second attempt at an LP solution.

9 Example

This example solves the integer programming problem:

maximize

F (x) = 3 x_{1} + 4 x_{2}

subject to the bounds

\begin{matrix} x_{1} \geq 0 \\ x_{2} \geq 0 \end{matrix}

and to the general constraints

\begin{array}{l} 2 x_{1} + 5 x_{2} \leq 15 \\ 2 x_{1} - 2 x_{2} \leq 5 \\ 3 x_{1} + 2 x_{2} \geq 5 \end{array}

where

x_{1}

and

x_{2}

are integer variables.

The initial point, which is feasible, is

x_{0} = {(1, 1)}^{T},

and

F (x_{0}) = 7

The optimal solution is

x^{*} = {(2, 2)}^{T},

and

F (x^{*}) = 14

Note that maximizing

F (x)

is equivalent to minimizing

- F (x)

NAG Library Routine DocumentH02BBF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

5.1 Description of Printed Output

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

H02BBF